What Is the Graph of max(|x|, |y|, |z|) = 1?

The equation $\max(|x|,|y|,|z|)=1$ describes a surface in three-dimensional space. The maximum of the three absolute coordinate values is 1, so no coordinate can be larger than 1 in magnitude, and at least one coordinate must have magnitude exactly 1.

That makes the graph the boundary of a cube centered at the origin, with side length 2. The cube extends from −1 to 1 along each coordinate axis.

Why the graph is a cube surface

For the maximum to equal 1, all three absolute values must be at most 1:

$$|x|\leq 1,\qquad |y|\leq 1,\qquad |z|\leq 1.$$

Also, at least one of those values must equal 1. For example, if $|x|=1$, then either $x=1$ or $x=-1$, while $y$ and $z$ may vary between −1 and 1. Those possibilities give two square faces. Applying the same reasoning to $y$ and $z$ gives the other four faces.

The six face descriptions are:

• $x=1$ and $x=-1$, with $-1\leq y,z\leq 1$;
• $y=1$ and $y=-1$, with $-1\leq x,z\leq 1$;
• $z=1$ and $z=-1$, with $-1\leq x,y\leq 1$.

These faces meet along the cube’s edges. The eight vertices are the points $(\pm1,\pm1,\pm1)$, meaning every combination of signs. The origin is inside the cube, not on the graph, because its maximum absolute coordinate is 0.

A quick test for points

To decide whether a point is on the surface, take the absolute value of each coordinate and find the largest.

For $(0.4,-1,0.7)$, the absolute values are $0.4$, $1$, and $0.7$. Their maximum is 1, so the point lies on the face $y=-1$.

For $(1,1,0)$, the maximum is also 1. This point lies on an edge where the faces $x=1$ and $y=1$ meet. For $(0.4,-0.8,0.7)$, the maximum is $0.8$, so the point is inside the cube but not on its surface. For $(1.2,0,0)$, the maximum is $1.2$, so the point is outside the cube.

How to sketch it

Draw the usual perspective outline of a cube centered at the origin. Mark the coordinate axes so their positive and negative directions reach the faces at $1$ and $-1$. For instance, the positive $x$-axis meets the face $x=1$ at $(1,0,0)$, and the negative $x$-axis meets the opposite face at $(-1,0,0)$. Do the same for the $y$- and $z$-axes.

The equation specifies the entire surface, not just the six face interiors: include the edges and vertices where faces meet. If you are using graphing software, graph the six plane pieces listed above, restricting the other two coordinates on each piece to the interval $[-1,1]$.

Related inequality: the solid cube

Changing the equality to

$$\max(|x|,|y|,|z|)\leq 1$$

includes every point inside the cube as well as its boundary. The strict inequality $\max(|x|,|y|,|z|)<1$ describes only the cube’s interior. By contrast, the original equality picks out the boundary surface alone.

This is different from $x^2+y^2+z^2=1$, which describes a sphere of radius 1. The maximum-coordinate condition measures the largest distance along any coordinate direction, so its level surface has flat square faces rather than a curved surface.

Common mistakes

• Drawing a sphere: The absolute values are not squared and added. A maximum of coordinate magnitudes produces a cube, not a sphere.
• Including the interior: Points where all three absolute coordinates are less than 1 do not satisfy the equality. They belong to the solid described by the corresponding inequality.
• Forgetting negative faces: Absolute values make both positive and negative coordinate directions count. There are two faces perpendicular to each axis, for six faces total.
• Leaving out edges: Points can have two or even three coordinates with absolute value 1. They still satisfy the equation and form the cube’s edges and vertices.